On Bi-R-Diagonal Pairs of Operators

Abstract

We study the properties of the analogue of R-diagonal operators in the setting of bi-free probability. Products of bi-R-diagonal pairs of operators that are $\ast$-bi-free are studied and powers of such pairs are found to also be bi-R-diagonal. It is moreover shown that the joint $\ast$-distribution of a bi-R-diagonal pair of operators remains invariant under the multiplication by a $\ast$-bi-free bi-Haar unitary pair and equivalent characterizations of bi-R-diagonal pairs are developed.

40 pages. v2: Major revisions. Paper significantly reorganized and expanded. New combinatorial framework introduced to streamline arguments, many proofs rewritten with added diagrams and several new results included (notably on distinct powers and independence of bi-R-diagonal pairs)

References in corpus (2)

• Combinatorics of Bi-Freeness with Amalgamation
• An alternating moment condition for bi-freeness

Cited by in corpus (1)

• Limit Theorems and Wrapping Transforms in Bi-free Probability Theory